Quantitative measurements using multiple frequency atomic force microscopy

ABSTRACT

The imaging mode presented here combines the features and benefits of amplitude modulated (AM) atomic force microscopy (AFM), sometimes called AC mode AFM, with frequency modulated (FM) AFM. In AM-FM imaging, the topographic feedback from the first resonant drive frequency operates in AM mode while the second resonant drive frequency operates in FM mode and is adjusted to keep the phase at 90 degrees, on resonance. With this approach, frequency feedback on the second resonant mode and topographic feedback on the first are decoupled, allowing much more stable, robust operation.

This application is a continuation of Ser. No. 13/694,095, filed Oct.29, 2012, now U.S. Pat. No. 9,297,827, issued Mar. 29, 2016, which was acontinuation-in-part application of U.S. Ser. No. 13/241,689 filed Sep.23, 2011, now U.S. Pat. No. 8,555,711 issued Sep. 27, 2011; which is acontinuation application of U.S. Ser. No. 12/214,031 filed Jun. 16,2008, now U.S. Pat. No. 8,024,963 issued Sep. 27, 2011, entitled“Quantitative Measurements Using Multiple Frequency Atomic ForceMicroscopy”. This application also claims priority from provisionalapplication No. 61/628,323, filed Oct. 27, 2011. The disclosures of eachof these parent applications are hereby incorporated by reference, intheir entirety.

BACKGROUND OF THE INVENTION

For the sake of convenience, the current description focuses on systemsand techniques that may be realized in a particular embodiment ofcantilever-based instruments, the atomic force microscope (AFM).Cantilever-based instruments include such instruments as AFMs, molecularforce probe instruments (1 D or 3D), high-resolution profilometers(including mechanical stylus profilometers), surface modificationinstruments, chemical or biological sensing probes, and micro-actuateddevices. The systems and techniques described herein may be realized insuch other cantilever-based instruments.

An AFM is a device used to produce images of surface topography (and/orother sample characteristics) based on information obtained fromscanning (e.g., rastering) a sharp probe on the end of a cantileverrelative to the surface of the sample. Topographical and/or otherfeatures of the surface are detected by detecting changes in deflectionand/or oscillation characteristics of the cantilever (e.g., by detectingsmall changes in deflection, phase, frequency, etc., and using feedbackto return the system to a reference state). By scanning the proberelative to the sample, a “map” of the sample topography or other samplecharacteristics may be obtained.

Changes in deflection or in oscillation of the cantilever are typicallydetected by an optical lever arrangement whereby a light beam isdirected onto the cantilever in the same reference frame as the opticallever. The beam reflected from the cantilever illuminates a positionsensitive detector (PSD). As the deflection or oscillation of thecantilever changes, the position of the reflected spot on the PSDchanges, causing a change in the output from the PSD. Changes in thedeflection or oscillation of the cantilever are typically made totrigger a change in the vertical position of the cantilever baserelative to the sample (referred to herein as a change in the Zposition, where Z is generally orthogonal to the XY plane defined by thesample), in order to maintain the deflection or oscillation at aconstant pre-set value. It is this feedback that is typically used togenerate an AFM image.

AFMs can be operated in a number of different sample characterizationmodes, including contact mode where the tip of the cantilever is inconstant contact with the sample surface, and AC modes where the tipmakes no contact or only intermittent contact with the surface.

Actuators are commonly used in AFMs, for example to raster the probe orto change the position of the cantilever base relative to the samplesurface. The purpose of actuators is to provide relative movementbetween different parts of the AFM; for example, between the probe andthe sample. For different purposes and different results, it may beuseful to actuate the sample, the cantilever or the tip or somecombination of both. Sensors are also commonly used in AFMs. They areused to detect movement, position, or other attributes of variouscomponents of the AFM, including movement created by actuators.

For the purposes of the specification, unless otherwise specified, theterm “actuator” refers to a broad array of devices that convert inputsignals into physical motion, including piezo activated flexures, piezotubes, piezo stacks, blocks, bimorphs, unimorphs, linear motors,electrostrictive actuators, electrostatic motors, capacitive motors,voice coil actuators and magnetostrictive actuators, and the term“position sensor” or “sensor” refers to a device that converts aphysical parameter such as displacement, velocity or acceleration intoone or more signals such as an electrical signal, including capacitivesensors, inductive sensors (including eddy current sensors),differential transformers (such as described in co-pending applicationsUS20020175677A1 and US20040075428A1, Linear Variable DifferentialTransformers for High Precision Position Measurements, andUS20040056653A1, Linear Variable Differential Transformer with DigitalElectronics, which are hereby incorporated by reference in theirentirety), variable reluctance, optical interferometry, opticaldeflection detectors (including those referred to above as a PSD andthose described in co-pending applications US20030209060A1 andUS20040079142A1, Apparatus and Method for Isolating and MeasuringMovement in Metrology Apparatus, which are hereby incorporated byreference in their entirety), strain gages, piezo sensors,magnetostrictive and electrostrictive sensors.

In both the contact and AC sample-characterization modes, theinteraction between the probe and the sample surface induces adiscernable effect on a probe-based operational parameter, such as thecantilever deflection, the cantilever oscillation amplitude, the phaseof the cantilever oscillation relative to the drive signal driving theoscillation or the frequency of the cantilever oscillation, all of whichare detectable by a sensor. In this regard, the resultantsensor-generated signal is used as a feedback control signal for the Zactuator to maintain a designated probe-based operational parameterconstant.

In contact mode, the designated parameter may be cantilever deflection.In AC modes, the designated parameter may be oscillation amplitude,phase or frequency. The feedback signal also provides a measurement ofthe sample characteristic of interest. For example, when the designatedparameter in an AC mode is oscillation amplitude, the feedback signalmay be used to maintain the amplitude of cantilever oscillation constantto measure changes in the height of the sample surface or other samplecharacteristics.

The periodic interactions between the tip and sample in AC modes inducescantilever flexural motion at higher frequencies. Measuring the motionallows interactions between the tip and sample to be explored. A varietyof tip and sample mechanical properties including conservative anddissipative interactions may be explored. Stark, et al., have pioneeredanalyzing the flexural response of a cantilever at higher frequencies asnonlinear interactions between the tip and the sample. In theirexperiments, they explored the amplitude and phase at numerous higheroscillation frequencies and related these signals to the mechanicalproperties of the sample.

Unlike the plucked guitar strings of elementary physics classes,cantilevers normally do not have higher oscillation frequencies thatfall on harmonics of the fundamental frequency. The first three modes ofa simple diving board cantilever, for example, are at the fundamentalresonant frequency (f₀), 6.19f₀ and 17.5 f₀. An introductory text incantilever mechanics such as Sarid has many more details. Throughcareful engineering of cantilever mass distributions, Sahin, et al.,have developed a class of cantilevers whose higher modes do fall onhigher harmonics of the fundamental resonant frequency. By doing this,they have observed that cantilevers driven at the fundamental exhibitenhanced contrast, based on their simulations on mechanical propertiesof the sample surface. This approach is has the disadvantage ofrequiring costly and difficult to manufacture special cantilevers.

The simple harmonic oscillator (SHO) model gives a convenientdescription at the limit of the steady state amplitude A of theeigenmode of a cantilever oscillating in an AC mode:

$A = {\frac{F_{0}/m}{\sqrt{\left( {\omega_{0}^{2} - \omega} \right)^{2} - \left( {{\omega\omega}_{0}/Q} \right)^{2}}}({SHO})}$where F₀ is the drive amplitude (typically at the base of thecantilever), m is the mass, ω is the drive frequency in units ofrad/sec, ω₀ is the resonant frequency and Q is the “quality” factor, ameasure of the damping.

If, as is often the case, the cantilever is driven through excitationsat its base, the expression becomes

$A = {\frac{A_{drive}\omega_{0}^{2}}{\sqrt{\left( {\omega_{0}^{2} - \omega^{2}} \right)^{2}\left( {\omega_{0}\omega} \right)^{2}}}\left( {{SHO}\mspace{14mu}{Amp}} \right)}$where F₀/m has been replaced with A_(drive)ω₀ ², where A_(drive) is thedrive amplitude (at the oscillator).

The phase angle φ is described by an associated equation

$\phi = {{\tan^{- 1}\left\lbrack \frac{{\omega\omega}_{0}}{Q\left( {\omega_{0}^{2} - \omega^{2}} \right)} \right\rbrack}\left( {{SHO}\mspace{14mu}{Phase}} \right)}$

When these equations are fulfilled, the amplitude and phase of thecantilever are completely determined by the user's choice of the drivefrequency and three independent parameters: A_(drive), ω₀ and Q.

In some very early work, Martin, et al., drove the cantilever at twofrequencies. The cantilever response at the lower, non-resonantfrequency was used as a feedback signal to control the surface trackingand produced a topographic image of the surface. The response at thehigher frequency was used to characterize what the authors interpretedas differences in the non-contact forces above the Si and photo-resiston a patterned sample.

Recently, Rodriguez and Garcia published a theoretical simulation of anon-contact, attractive mode technique where the cantilever was drivenat its two lowest eigenfrequencies. In their simulations, they observedthat the phase of the second mode had a strong dependence on the Hamakerconstant of the material being imaged, implying that this techniquecould be used to extract chemical information about the surfaces beingimaged. Crittenden et al. have explored using higher harmonics forsimilar purposes.

There are a number of techniques where the instrument is operated in ahybrid mode where a contact mode feedback loop is maintained while someparameter is modulated. Examples include force modulation andpiezo-response imaging.

Force modulation involves maintaining a contact mode feedback loop whilealso driving the cantilever at a frequency and then measuring itsresponse. When the cantilever makes contact with the surface of thesample while being so driven, its resonant behavior changessignificantly. The resonant frequency typically increases, depending onthe details of the contact mechanics. In any event, one may learn moreabout the surface properties because the elastic response of the samplesurface is sensitive to force modulation. In particular, dissipativeinteractions may be measured by measuring the phase of the cantileverresponse with respect to the drive.

A well-known shortcoming of force modulation and other contact modetechniques is that the while the contact forces may be controlled well,other factors affecting the measurement may render it ill-defined. Inparticular, the contact area of the tip with the sample, usuallyreferred to as contact stiffness, may vary greatly depending on tip andsample properties. This in turn means that the change in resonance whilemaintaining a contact mode feedback loop, which may be called thecontact resonance, is ill-defined. It varies depending on the contactstiffness. This problem has resulted in prior art techniques avoidingoperation at or near resonance.

SUMMARY OF THE INVENTION

Cantilevers are continuous flexural members with a continuum ofvibrational modes. The present invention describes different apparatusand methods for exciting the cantilever simultaneously at two or moredifferent frequencies and the useful information revealed in the imagesand measurements resulting from such methods. Often, these frequencieswill be at or near two or more of the cantilever vibrational eigenmodes

Past work with AC mode AFMs has been concerned with higher vibrationalmodes in the cantilever, with linear interactions between the tip andthe sample. The present invention, however, is centered aroundnon-linear interactions between the tip and sample that couple energybetween two or more different cantilever vibrational modes, usually keptseparate in the case of linear interactions. Observing the response ofthe cantilever at two or more different vibrational modes has someadvantages in the case of even purely linear interactions however. Forexample, if the cantilever is interacting with a sample that has somefrequency dependent property, this may show itself as a difference inthe mechanical response of the cantilever at the different vibrationalmodes.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 Preferred embodiment for probing multiple eigenmodes of acantilever.

FIG. 2 Preferred embodiment for exciting voltage-dependent motion in thecantilever probe.

FIG. 3 Preferred embodiment for probing an active device.

FIG. 4 Phase and amplitude shifts of the fundamental eigenmode with andwithout the second eigenmode being driven.

FIG. 5A-5E Images of collagen fibers taken with the preferredembodiment.

FIGS. 6 and 7 Two dimensional histogram plots of the amplitude and phasefor the first and second eigenmodes.

FIG. 8 Preferred embodiment for probing an active sample in contactwhile measuring dynamic contact properties (Dual Frequency ResonanceTracking Piezo Force Microscopy (DFRT PFM)).

FIG. 9 Resonance peaks in sweep of applied potential from dc to 2 MHz.

FIG. 10A-10D and 11 Images of a piezoelectric sample when the cantileverpotential was driven at two different frequencies, one slightly belowand the other slightly above the same contact resonance frequency.

FIG. 12 Amplitude versus frequency and phase versus frequency curvessimultaneously measured at different frequencies.

FIG. 13 Amplitude and phase curves changing in response to varyingtip-sample interactions being driven first at two different frequenciesand then at a single frequency.

FIG. 14 Amplitude versus frequency sweeps around the second resonancemade while feeding back on the first mode amplitude.

FIG. 15-16 Amplitude versus frequency and phase versus frequency curvessimultaneous measured at different frequencies.

FIG. 17-19 Images of a piezoelectric sample when the cantileverpotential was driven at two different frequencies, one slightly belowand the other slightly above the same contact resonance frequency.

FIG. 20 Preferred embodiment of an apparatus for probing the first twoflexural resonances of a cantilever and imaging in AM mode with phase isand FM mode in accordance with the present invention.

FIG. 21 Topography of a Si-epoxy (SU8) patterned wafer imaged using theLoss Tangent technique of the present invention.

FIG. 22 Steps in calculating the corrected Loss Tangent.

FIG. 23 Direct measurement of the second mode tip-sample interactionforces.

FIG. 24 EPDH/Epoxy cryo-microtomed boundary measured at 2 Hz and 20 Hzline scan rates.

FIG. 25 Simultaneous mapping of loss tangent and stiffness of anelastomer-epoxy sandwich.

FIG. 26 Simplified measurement of the phase of the second mode for smallfrequency shifts.

FIG. 27 Effects of choosing resonant modes that are softer, matched orstiffer than the tip-sample stiffness,

FIG. 28 Extension of thermal noise measurement method to higher modes.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 is a block diagram of a preferred embodiment of an apparatus forprobing multiple eigenmodes of a cantilever in accordance with thepresent invention. The sample 1010 is positioned below the cantileverprobe 1020. The chip of the cantilever probe 1030 is driven by amechanical actuator 1040, preferably a piezoelectric actuator, but othermethods to induce cantilever motion known to those versed in the artcould also be used. The motion of the cantilever probe 1020 relative tothe frame of the microscope 1050 is measured with a detector 1060, whichcould be an optical lever or another method known to those versed in theart. The cantilever chip 1030 is moved relative to the sample 1010 by ascanning apparatus 1070, preferably a piezo/flexure combination, butother methods known to those versed in the art could also be used.

The motion imparted to the cantilever chip 1030 by actuator 1040 iscontrolled by excitation electronics that include at least two frequencysynthesizers 1080 and 1090. There could be additional synthesizers ifmore than two cantilever eigenmodes are to be employed. The signals fromthese frequency synthesizers could be summed together by an analogcircuit element 1100 or, preferably, a digital circuit element thatperforms the same function. The two frequency synthesizers 1080 and 1090provide reference signals to lockin amplifiers 1110 and 1120,respectively. In the case where more than two eigenmodes are to beemployed, the number of lockin amplifiers will also be increased. Aswith other electronic components in this apparatus, the lockinamplifiers 1110 and 1120 can be made with analog circuitry or withdigital circuitry or a hybrid of both. For a digital lockin amplifier,one interesting and attractive feature is that the lockin analysis canbe performed on the same data stream for both eigenmodes. This impliesthat the same position sensitive detector and analog to digitalconverter can be used to extract information at the two distincteigenmodes.

The lockin amplifiers could also be replaced with rms measurementcircuitry where the rms amplitude of the cantilever oscillation is usedas a feedback signal.

There are a number of variations in the FIG. 1 apparatus that a personskilled in the art could use to extract information relative to thedifferent eigenmodes employed in the present invention. Preferably, adirect digital synthesizer (DDS) could be used to create sine and cosinequadrature pairs of oscillating voltages, each at a frequency matched tothe eigenmodes of the cantilever probe 1030 that are of interest. Thisimplementation also allows dc voltages to be applied, allowing methodssuch as scanning Kelvin probing or simultaneous current measurementsbetween the tip and the sample. The amplitude and phase of eacheigenmode can be measured and used in a feedback loop calculated by thecontroller 1130 or simply reported to the user interface 1140 where itis displayed, stored and/or processed further in an off-line manner.Instead of, or in addition to, the amplitude and phase of the cantilevermotion, the quadrature pairs, usually designated x and y, can becalculated and used in a manner similar to the amplitude and phase.

In one method of using the FIG. 1 apparatus, the cantilever is driven ator near two or more resonances by the single “shake” piezo 1040.Operating in a manner similar to AC mode where the cantilever amplitudeis maintained constant and used as a feedback signal, but employing theteachings of the present invention, there are now a number of choicesfor the feedback loop. Although the work here will focus on using theamplitude of the fundamental (A₀), we were able to successfully imageusing one of the higher mode amplitudes (A₁) as a feedback signal aswell as a sum of all the amplitudes A₀+A₁+ . . . . One can also chooseto exclude one or more modes from such a sum. So for example, wherethree modes are employed, the sum of the first and second could be usedto operate the feedback loop and the third could be used as a carryalong signal.

Because higher eigenmodes have a significantly higher dynamic stiffness,the energy of these modes can be much larger that that of lowereigenmodes.

The method may be used to operate the apparatus with one flexural modeexperiencing a net attractive force and the other a net repulsive force,as well as operating with each mode experiencing the same net sign offorce, attractive or repulsive. Using this method, with the cantileverexperiencing attractive and repulsive interactions in differenteigenmodes, may provide additional information about sample properties.

One preferred technique for using the aforesaid method is as follows.First, excite the probe tip at or near a resonant frequency of thecantilever keeping the tip sufficiently far from the sample surface thatit oscillates at the free amplitude A₁₀ unaffected by the proximity ofthe cantilever to the sample surface and without making contact with thesample surface. At this stage, the cantilever is not touching thesurface; it turns around before it interacts with significant repulsiveforces.

Second, reduce the relative distance in the Z direction between the baseof the cantilever and the sample surface so that the amplitude of theprobe tip A₁ is affected by the proximity of the sample surface withoutthe probe tip making contact with the sample surface. The phase p₁ willbe greater than p₁₀, the free first eigenmode phase. This amplitude ismaintained at an essentially constant value during scanning without theprobe tip making contact with the sample surface by setting up afeedback loop that controls the distance between the base of thecantilever and the sample surface.

Third, keeping the first eigenmode drive and surface controllingfeedback loop with the same values, excite a second eigenmode of thecantilever at an amplitude A₂. Increase A₂ until the second eigenmodephase p₂ shows that the cantilever eigenmode is interacting withpredominantly repulsive forces; that is, that p₂ is less than p₂₀, thefree second eigenmode phase. This second amplitude A₂ is not included inthe feedback loop and is allowed to freely roam over a large range ofvalues. In fact, it is typically better if variations in A₂ can be aslarge as possible, ranging from 0 to A₂₀, the free second eigenmodeamplitude.

Fourth, the feedback amplitude and phase, A₁ and p₁, respectively, aswell as the carry along second eigenmode amplitude and phase, A₂ and p₂,respectively, should be measured and displayed.

Alternatively, the drive amplitude and/or phase of the second frequencycan be continually adjusted to maintain the second amplitude and/orphase at an essentially constant value. In this case, it is useful todisplay and record the drive amplitude and/or frequency required tomaintain the second amplitude and/or phase at an essentially constantvalue.

A second preferred technique for using the aforesaid method follows thefirst two steps of first preferred technique just described and thencontinues with the following two steps:

Third, keeping the first eigenmode drive and surface controllingfeedback loop with the same values, excite a second eigenmode (orharmonic) of the cantilever at an amplitude A₂. Increase A₂ until thesecond eigenmode phase p₂ shows that the cantilever eigenmode isinteracting with predominantly repulsive forces; that is, that p₂ isless than p₂₀, the free second eigenmode phase. At this point, thesecond eigenmode amplitude A₂ should be adjusted so that the firsteigenmode phase p₁ becomes predominantly less than p₁₀, the free firsteigenmode phase. In this case, the adjustment of the second eigenmodeamplitude A₂ has induced the first eigenmode of the cantilever tointeract with the surface in a repulsive manner. As with the firstpreferred technique, the second eigenmode amplitude A₂ is not used inthe tip-surface distance feedback loop and should be allowed rangewidely over many values.

Fourth, the feedback amplitude and phase, A₁ and p₁, respectively, aswell as the carry along second eigenmode amplitude and phase, A₂ and p₂,respectively, should be measured and displayed.

Either of the preferred techniques just described could be performed ina second method of using the FIG. 1 apparatus where the phase of theoscillating cantilever is used in a feedback loop and the oscillationfrequency is varied to maintain phase essentially constant. In thiscase, it is preferable to use the oscillation frequency as an input intoa z-feedback loop that controls the cantilever-sample separation.

Relative changes in various parameters such as the amplitude and phaseor in-phase and quadrature components of the cantilever at thesedifferent frequencies could also be used to extract information aboutthe sample properties.

A third preferred technique for using the first method of using the FIG.1 apparatus provides an alternative for conventional operation in arepulsive mode, that is where the tip is experiencing a net repulsiveforce. The conventional approach for so operating would be to use alarge amplitude in combination with a lower setpoint, and a cantileverwith a very sharp tip. Using this third preferred technique, however,the operator begins, just as with the first two techniques, by choosingan amplitude and setpoint for the fundamental eigenmode that is smallenough to guarantee that the cantilever is experiencing attractiveforces, that is, that the cantilever is in non-contact mode. As notedbefore, this operational mode can be identified by observing the phaseof the cantilever oscillation. In the non-contact case, the phase shiftis positive, implying that the resonant frequency has been lowered. Withthese conditions on the first eigenmode, the second eigenmode excitationcan be introduced and the amplitude, drive frequency and, if applicable,set point chosen with the following considerations in mind:

1. Both eigenmodes are in the attractive mode, that is to say that thephase shift of both modes is positive, implying both eigenmodefrequencies have been shifted negatively by the tip-sample interactions.Generally, this requires a small amplitude for the second eigenmode.

2. The fundamental eigenmode remains attractive while the secondeigenmode is in a state where the tip-sample interactions cause it to bein both the attractive and the repulsive modes as it is positionedrelative to the surface.

3. The fundamental eigenmode is in an attractive mode and the secondeiegenmode is in a repulsive mode.

4. In the absence of any second mode excitation, the first eigenmode isinteracting with the surface in the attractive mode. After the secondeigenmode is excited, the first eigenmode is in a repulsive mode. Thischange is induced by the addition of the second eigenmode energy. Thesecond eigenmode is in a state where the tip-sample interactions causeit to be attractive and/or repulsive.

5. The first eigenmode is in a repulsive mode and the second mode is ina repulsive mode.

The transition from attractive to repulsive mode in the first eigenmode,as induced by the second eigenmode excitation, is illustrated in FIG. 4,where the amplitude and phase of the first and second eigenmodes areplotted as a function of the distance between the base of the cantileverand the surface of the sample. The point where the tip begins tointeract significantly with the surface is indicated with a solid line4000. The fundamental amplitude 4010 of the cantilever decreases as thecantilever starts to interact with the surface, denoted by the solidline 4000. The associated phase 4020 shows a positive shift, consistentwith overall attractive interactions. For these curves, the secondeigenmode amplitude is zero and therefore not plotted in the Figure (andneither is phase, for the same reason). Next, the second eigenmode isexcited and the same curves are re-measured, together with the amplitudeand phase of the second eigenmode, 4030 and 4040. There is a notablechange in the fundamental eigenmode amplitude 4050 and more strikingly,the fundamental eigenmode phase 4060. The fundamental phase in factshows a brief positive excursion, but then transitions to a negativephase shift, indicating an overall repulsive interaction between the tipand sample. The free amplitude of the first eigenmode is identical inboth cases, the only difference in the measurement being the addition ofenergy exciting the higher oscillatory eigenmode. This excitation issufficient to drive the fundamental eigenmode into repulsiveinteractions with the sample surface. Furthermore, the phase curve ofthe second eigenmode indicates that it is also interacting overallrepulsively with the sample surface.

More complicated feedback schemes can also be envisioned. For example,one of the eigenmode signals can be used for topographical feedbackwhile the other signals could be used in other feedback loops. Anexample would be that A₁ is used to control the tip-sample separationwhile a separate feedback loop was used to keep A₂ at an essentiallyconstant value rather than allowing it to range freely over many values.A similar feedback loop could be used to keep the phase of the secondfrequency drive p₂ at a predetermined value with or without the feedbackloop on A₂ being implemented.

As another example of yet another type of feedback that could be used,Q-control can also be used in connection with any of the techniques forusing the first method of using the FIG. 1 apparatus. Using Q-control onany or all of the eigenmodes employed can enhance their sensitivity tothe tip-sample forces and therefore mechanical or other properties ofthe sample. It can also be used to change the response time of theeigenmodes employed which may be advantageous for more rapidly imaging asample. For example, the value of Q for one eigenmode could be increasedand the value for another decreased. This may enhance the result ofmixed attractive/repulsive mode imaging because it is generally easierto keep one eignmode interacting with the sample in repulsive mode witha reduced Q-value or, conversely, in attractive mode with an enhancedQ-value. By reducing the Q-value of the lowest eigenmode and enhancingthe Q-value of the next eigenmode, it is possible to encourage the mixedmode operation of the cantilever; the zeroth eigenmode will be inrepulsive mode while the first eigenmode will more likely remain inattractive mode. Q-control can be implemented using analog, digital orhybrid analog-digital electronics. It can be accomplished using anintegrated control system such as that in the Asylum ResearchCorporation MFP-3D Controller or by after-market modules such as thenanoAnalytics Q-box.

In addition to driving the cantilever at or near more than oneeigenmode, it is possible to also excite the cantilever at or near oneor more harmonics and/or one or more eigenmodes. It has been known forsome time that nonlinear interactions between the tip and the sample cantransfer energy into cantilever harmonics. In some cases this energytransfer can be large but it is usually quite small, on the order of apercent of less of the energy in the eigenmode. Because of this, theamplitude of motion at a harmonic, even in the presence of significantnonlinear coupling is usually quite small. Using the methods describedhere, it is possible to enhance the contrast of these harmonics bydirectly driving the cantilever at the frequency of the harmonic. Tofurther enhance the contrast of this imaging technique it is useful toadjust the phase of the higher frequency drive relative to that of thelower. This method improves the contrast of both conventionalcantilevers and the specially engineered “harmonic” cantileversdescribed by Sahin et al and other researchers.

On many samples, the results of imaging with the present invention aresimilar to, and in some cases superior to, the results of conventionalphase imaging. However, while phase imaging often requires a judiciouschoice of setpoint and drive amplitude to maximize the phase contrast,the method of the present invention exhibits high contrast over a muchwider range of imaging parameters. Moreover, the method also works influid and vacuum, as well as air and the higher flexural modes showunexpected and intriguing contrast in those environments, even onsamples such as DNA and cells that have been imaged numerous timesbefore using more conventional techniques.

Although there is a wide range of operating parameters that yieldinteresting and useful data, there are situations where more carefultuning of the operational parameters will yield enhanced results. Someof these are discussed below. Of particular interest can be regions inset point and drive amplitude space where there is a transition fromattractive to repulsive (or vice versa) interactions in one or more ofthe cantilever eigenmodes or harmonics.

The superior results of imaging with the present invention may be seenfrom an inspection of the images. An example is shown in FIG. 5. Forthis example, the FIG. 1 apparatus was operated using the fundamentaleignemode amplitude as the error signal and the second eigenmode as acarry-along signal. The topography image 5010 in FIG. 5 shows collagenfibers on a glass surface, an image typical of results with conventionalAC mode from similar samples. The fundamental eigenmode amplitude image5020 is relatively similar, consistent with the fundamental eignemodeamplitude being used in the feedback loop. The fundamental eigenmodephase channel image 5030 shows some contrast corresponding to edges inthe topography image. This is consistent with the interaction being moreattractive at these regions, again to be expected from surface energyconsiderations (larger areas in proximity will have larger long-rangeattractive forces). Since the fundamental eignemode amplitude is beingheld relatively constant and there is a relationship between theamplitude and phase, the phase will be constrained, subject to energybalance and the feedback loop that is operating to keep the amplitudeconstant. The second eigenmode amplitude image 5040 shows contrast thatis similar to the fundamental eigenmode phase image 5030. However, thereare some differences, especially over regions thought to be contaminants5041 and 5042. Finally, the second eigenmode phase image 5050 shows themost surprisingly large amount of contrast. The background substrate5053 shows a bright, positive phase contrast. The putative contaminantpatches, 5041, 5042 and 5051 show strikingly dark, negative phase shiftcontrast. Finally, regions where the collagen fibers are suspended 5052show dark, negative phase contrast. In these last regions, the suspendedcollagen fibers are presumably absorbing some of the vibrational energyof the second eigenmode amplitude and thus, changing the response.

When an AFM is operated in conventional amplitude modulated (AM) AC modewith phase detection, the cantilever amplitude is maintained constantand used as a feedback signal. Accordingly, the values of the signalused in the loop are constrained not only by energy balance but also bythe feedback loop itself. Furthermore, if the amplitude of thecantilever is constrained, the phase will also be constrained, subjectto conditions discussed below. In conventional AC mode it is not unusualfor the amplitude to vary by a very small amount, depending on the gainsof the loop. This means that, even if there are mechanical properties ofthe sample that might lead to increased dissipation or a frequency shiftof the cantilever, the z-feedback loop in part corrects for thesechanges and thus in this sense, avoids presenting them to the user.

If the technique for using the present invention involves a mode that isexcited but not used in the feedback loop, there will be no explicitconstraints on the behavior of this mode. Instead it will range freelyover many values of the amplitude and phase, constrained only by energybalance. That is to say, the energy that is used to excite thecantilever motion must be balanced by the energy lost to the tip-sampleinteractions and the intrinsic dissipation of the cantilever. This mayexplain the enhanced contrast we observe in images generated with thetechniques of the present invention.

FIG. 6 demonstrates this idea more explicitly. The first image 6010 isan image of the number of pixels at different amplitudes (horizontalaxis) and phases (vertical axis) in the fundamental eigenmode data forthe collagen sample of FIG. 5. As expected, the amplitude values areconstrained to a narrow range around ˜0.6Amax by the z-feedback loop.Constraining the amplitude values in turn, limits the values that thephase can take to the narrow range around 25°. Thus, when the pixelcounts are plotted, there is a bright spot 6020 with only smallvariations. Small variations in turn imply limited contrast. The secondimage 6030 plots the number of pixels at different amplitudes and phasesin the second eigenmode data for the collagen sample. Since theamplitude of this eigenmode was not constrained by a feedback loop, itvaries from Amax to close to zero. Similarly, the phase ranges over manyvalues. This freedom allows greatly increased contrast in the secondeigenmode images.

The present invention may also be used in apparatus that induce motionin the cantilever other than through a piezoelectric actuator. Thesecould include direct electric driving of the cantilever (“activecantilevers”), magnetic actuation schemes, ultrasonic excitations,scanning Kelvin probe and electrostatic actuation schemes.

Direct electric driving of the cantilever (“active cantilevers”) usingthe present invention has several advantages over conventional piezoforce microscopy (PFM) where the cantilever is generally scanned overthe sample in contact mode and the cantilever voltage is modulated in amanner to excite motion in the sample which in turn causes thecantilever to oscillate.

FIG. 2 is a block diagram of a preferred embodiment of an apparatus forusing the present invention with an active cantilever. This apparatushas similarities to that shown in FIG. 1, as well as differences. In theFIG. 2 apparatus, like the FIG. 1 apparatus, one of the frequencysources 1080 is used to excite motion of the cantilever probe 1020through a mechanical actuator 1040, preferably a piezoelectric actuator,but other methods to induce cantilever motion known to those versed inthe art could also be used, which drives the chip 1030 of the cantileverprobe 1020, However, in the FIG. 2 apparatus, the frequency source 1080communicates directly 2010 with the actuator 1040 instead of beingsummed together with the second frequency source 1090, as in the FIG. 1apparatus. The second frequency source 1090 in the FIG. 2 apparatus isused to vary the potential of the cantilever probe 1020 which in turncauses the sample 1010 to excite motion in the cantilever probe 1020 ata different eigenmode than that excited by the first frequency source1080. The resulting motion of the cantilever probe 1020 interacting withthe sample 1010 will contain information on the sample topography andother properties at the eigenmode excited by the first frequency source1080 and information regarding the voltage dependent properties of thesample at the eigenmode excited by the second frequency source 1090. Thesample holder 2030 can optionally be held at a potential, or at ground,to enhance the effect.

In one method of using the FIG. 2 apparatus, the amplitude of thecantilever at the frequency of the first source 1080 is used as theerror signal. The amplitude and phase (or in-phase and quadraturecomponents) at the frequency of the second source 1090 or a harmonicthereof will contain information about the motion of the sample andtherefore the voltage dependent properties of the sample. One example ofthese properties is the piezo-response of the sample. Another is theelectrical conductivity, charge or other properties that can result inlong range electrostatic forces between the tip and the sample.

FIG. 3 is a block diagram of a preferred embodiment of an apparatus forusing the present invention with the second frequency source modulatinga magnetic field that changes a property of the sample. In the FIG. 3apparatus, the situation with the first frequency source 1080 isidentical to the situation in the FIG. 2 apparatus. However, instead ofthe second frequency source 1090 being used to vary the potential of thecantilever probe 1020, as with the FIG. 2 apparatus, in the FIG. 3apparatus the second frequency source 1090 modulates the current throughan excitation coil 3010 which in turn modulates the magnetic state of amagnetic circuit element 3020. Magnetic circuit element 3020 could beused to modulate the field near an active sample or the excitation coil3010. Alternatively, magnetic circuit element 3020 could comprise thesample, as in the case of a magnetic recording head.

The FIG. 3 apparatus can be used with any other sort of ‘active’ samplewhere the interaction between the cantilever and the sample can bemodulated at or near one or more of the cantilever flexural resonancesby one of the frequency sources 1080 or 1090. This could also beextended to high frequency measurements such as described in Proksch etal., Appl. Phys. Lett., vol. (1999). Instead of the modulation describedin that paper, the envelope of the high frequency carrier could bedriven with a harmonic of one or more flexural resonances. This methodof measuring signals other than topographic has the advantage ofrequiring only one pass to complete as opposed to “LiftMode” or Nap modethat require temporally separated measurements of the topographic andother signals.

Another example of a preferred embodiment of an apparatus and method forusing the present invention is from the field of ultrasonic forcemicroscopy. In this embodiment, one or more eigenmodes are used for thez-feedback loop and one or more additional eigenmodes can be used tomeasure the high frequency properties of the sample. The high frequencycarrier is amplitude modulated and either used to drive the sampledirectly or to drive it using the cantilever as a waveguide. Thecantilever deflection provides a rectified measure of the sampleresponse at the carrier frequency.

Another group of embodiments for the present invention has similaritiesto the conventional force modulation technique described in theBackground to the Invention and conventional PFM where the cantilever isscanned over the sample in contact mode and a varying voltage is appliedto the cantilever. In general this group may be described as contactresonance embodiments. However, these embodiments, like the otherembodiments already described, make use of multiple excitation signals.

FIG. 8 is a block diagram of the first of these embodiments, which maybe referred to as Dual Frequency Resonance Tracking Piezo ForceMicroscopy (DFRT PFM). In the DFRT PFM apparatus of FIG. 8 thecantilever probe 1020 is positioned above a sample 1010 withpiezoelectric properties and scanned relative to the sample 1010 by ascanning apparatus 1070 using contact mode. Unlike conventional contactmode however the chip 1030 of the cantilever probe 1020, or thecantilever probe 1020 itself (alternative not shown), is driven byexcitation electronics that include at least two frequency synthesizers1080 and 1090. The cantilever probe 1020 responds to this excitation bybuckling up and down much as a plucked guitar string. The signals fromthese frequency synthesizers could be summed together by an analogcircuit element 1100 or, preferably, a digital circuit element thatperforms the same function. The two frequency synthesizers 1080 and 1090provide reference signals to lockin amplifiers 1110 and 1120,respectively. The motion of the cantilever probe 1020 relative to theframe of the microscope 1050 is measured with a detector 1060, whichcould be an optical lever or another method known to those versed in theart. The cantilever chip 1030 is moved vertically relative to the sample1010, in order to maintain constant force, by a scanning apparatus 1070,preferably a piezo/flexure combination, but other methods known to thoseversed in the art could also be used. The amplitude and phase of eachfrequency at which the cantilever probe 1020 is excited can be measuredand used in a feedback loop calculated by the controller 1130 or simplyreported to the user interface 1140 where it is displayed, stored and/orprocessed further in an off-line manner. Instead of, or in addition to,the amplitude and phase of the cantilever motion, the quadrature pairs,usually designated x and y, can be calculated and used in a mannersimilar to the amplitude and phase.

In one method of using the FIG. 8 apparatus, the topography of thesample would be measured in contact mode while the amplitude and phaseof the cantilever probe 1020 response to the applied potential at thelowest contact resonance and at the next highest contact resonance issimultaneously measured. The responses can be analyzed to determinewhether they originate from the actual piezoelectric response of thesample or from crosstalk between the topography and any electric forcesbetween the tip of the cantilever probe 1020 and the sample. Even moreinformation can be obtained if more frequencies are utilized.

FIG. 12 shows three examples of the changes in the native phase 12015and amplitude 12010 of a cantilever with a resonant frequency f₀ causedby interactions between the tip and the sample using DFRT PFM methods.These examples are a subset of changes that can be observed. In thefirst example, the resonant frequency is significantly lowered to f₀′but not damped. The phase 12085 and amplitude 12080 change but littlerelative to the native phase 12015 and amplitude 12010. In the secondexample the resonant frequency is again lowered to f₀′, this time withdamping of the amplitude. Here the phase 12095 is widened and theamplitude 12090 is appreciably flattened. Finally, in the third example,the resonant frequency is again dropped to f₀′, this time with areduction in the response amplitude. This yields a phase curve with anoffset 12105 but with the same width as the second case 12095 and areduced amplitude curve 12100 with the damping equivalent to that of thesecond example. If there is an offset in the phase versus frequencycurve as there is in this third example, prior art phase locked-loopelectronics will not maintain stable operation. For example, if thephase setpoint was made to be 90 degrees, it would never be possible tofind a frequency in curve 12105 where this condition was met. Oneexample of these things occurring in a practical situation is in DFRTPFM when the tip crosses from an electric domain with one orientation toa second domain with another orientation. The response induced by thesecond domain will typically have a phase offset with respect to thefirst. This is, in fact where the large contrast in DFRT PFM phasesignals originates.

FIG. 9 shows the cantilever response when the applied potential is sweptfrom dc to 2 MHz using the DFRT PFM apparatus. Three resonance peaks arevisible. Depending on the cantilever probe and the details of thetip-sample contact mechanics, the number, magnitude, breadth andfrequency of the peaks is subject to change. Sweeps such as these areuseful in choosing the operating points for imaging and othermeasurements. In a practical experiment, any or all of these resonancepeaks or the frequencies in between could be exploited by the methodssuggested above.

FIG. 19 shows a measurement that can be made using DFRT PFM techniques.A phase image 19010 shows ferroelectric domains written onto a sol-gelPZT surface. Because of the excellent separation between topography andPFM response possible with DFRT PFM, the phase image shows only piezoresponse, there is no topographic roughness coupling into the phase. Thewritten domains appear as bright regions. The writing was accomplishedby locally performing and measuring hysteresis loops by applying a dcbias to the tip during normal DFRT PFM operation. This allows the localswitching fields to be measured. The piezo phase 19030 and the amplitude19040 during a measurement made at location 19020 are plotted as afunction of the applied dc bias voltage. The loops were made followingStephen Jesse et al, Rev. Sci. Inst. 77, 073702 (2006). Other loops weretaken at the bright locations in image 19010, but are not shown in theFigure.

DFRT PFM is very stable over time in contrast to single frequencytechniques. This allows time dependent processes to be studied as isdemonstrated by the sequence of images, 19010, 19050, 19060, 19070 and19080 taken over the span of about 1.5 hours. In these images, thewritten domains are clearly shrinking over time.

In AC mode atomic force microscopy, relatively tiny tip-sampleinteractions can cause the motion of a cantilever probe oscillating atresonance to change, and with it the resonant frequency, phase,amplitude and deflection of the probe. Those changes of course are thebasis of the inferences that make AC mode so useful. With contactresonance techniques the contact between the tip and the sample also cancause the resonant frequency, phase and amplitude of the cantileverprobe to change dramatically.

The resonant frequency of the cantilever probe using contact resonancetechniques depends on the properties of the contact, particularly thecontact stiffness. Contact stiffness in turn is a function of the localmechanical properties of the tip and sample and the contact area. Ingeneral, all other mechanical properties being equal, increasing thecontact stiffness by increasing the contact area, will increase theresonant frequency of the oscillating cantilever probe. Thisinterdependence of the resonant properties of the oscillating cantileverprobe and the contact area represents a significant shortcoming ofcontact resonance techniques. It results in “topographical crosstalk”that leads to significant interpretational issues. For example, it isdifficult to know whether or not a phase or amplitude change of theprobe is due to some sample property of interest or simply to a changein the contact area.

The apparatus used in contact resonance techniques can also cause theresonant frequency, phase and amplitude of the cantilever probe tochange unpredictably. Examples are discussed by Rabe et al., Rev. Sci.Instr. 67, 3281 (1996) and others since then. One of the most difficultissues is that the means for holding the sample and the cantilever probeinvolve mechanical devices with complicated, frequency dependentresponses. Since these devices have their own resonances and damping,which are only rarely associated with the sample and tip interaction,they may cause artifacts in the data produced by the apparatus. Forexample, phase and amplitude shifts caused by the spurious instrumentalresonances may freely mix with the resonance and amplitude shifts thatoriginate with tip-sample interactions.

It is advantageous to track more than two resonant frequencies as theprobe scans over the surface when using contact resonance techniques.Increasing the number of frequencies tracked provides more informationand makes it possible to over-constrain the determination of variousphysical properties. As is well known in the art, this is advantageoussince multiple measurements will allow better determination of parametervalues and provide an estimation of errors.

Since the phase of the cantilever response is not a well behavedquantity for feedback purposes in PFM, we have developed other methodsfor measuring and/or tracking shifts in the resonant frequency of theprobe. One method is based on making amplitude measurements at more thanone frequency, both of which are at or near a resonant frequency. FIG.15 illustrates the idea. The original resonant frequency curve 14010 hasamplitudes A₁ 14030 and A₂ 14020, respectively, at the two drivefrequencies f₁ and f₂. However, if the resonant frequency shifted to alower value, the curve shifts to 14050 and the amplitudes at themeasurement frequencies change, A′₁ 14035 increasing and A′₂ 14025decreasing. If the resonant frequency were higher, the situation wouldreverse, that is the amplitude A′₁ at drive frequency f₁ would decreaseand A′₂ at f₂ would increase.

There are many methods to track the resonant frequency with informationon the response at more than one frequency. One method with DFRT PFM isto define an error signal that is the difference between the amplitudeat f₁ and the amplitude at f₂, that is A₁ minus A₂. A simpler examplewould be to run the feedback loop such that A₁ minus A₂=0, althoughother values could equally well be chosen. Alternatively both f₁ and f₂could be adjusted so that the error signal, the difference in theamplitudes, is maintained. The average of these frequencies (or evensimply one of them) provides the user with a measure of the contactresonant frequency and therefore the local contact stiffness. It is alsopossible to measure the damping and drive with the two values ofamplitude. When the resonant frequency has been tracked properly, thepeak amplitude is directly related to the amplitude on either side ofresonance. One convenient way to monitor this is to simply look at thesum of the two amplitudes. This provides a better signal to noisemeasurement than does only one of the amplitude measurements. Other,more complicated feedback loops could also be used to track the resonantfrequency. Examples include more complex functions of the measuredamplitudes, phases (or equivalently, the in-phase and quadraturecomponents), cantilever deflection or lateral and/or torsional motion.

The values of the two amplitudes also allow conclusions to be drawnabout damping and drive amplitudes. For example, in the case of constantdamping, an increase in the sum of the two amplitudes indicates anincrease in the drive amplitude while the difference indicates a shiftin the resonant frequency.

Finally, it is possible to modulate the drive amplitude and/orfrequencies and/or phases of one or more of the frequencies. Theresponse is used to decode the resonant frequency and, optionally,adjust it to follow changes induced by the tip-sample interactions.

FIG. 10 shows the results of a measurement of a piezo-electric materialusing DFRT PFM methods. Contact mode is used to image the sampletopography 10010 and contact resonance techniques used to image thefirst frequency amplitude 10020, the second frequency amplitude 10030,the first frequency phase 10040 and the second frequency phase 10050. Inthis experiment, the two frequencies were chosen to be close to thefirst contact resonance, at roughly the half-maximum point, with thefirst frequency on the lower side of the resonance curve and the secondon the upper side. This arrangement allowed some of the effects ofcrosstalk to be examined and potentially eliminated in subsequentimaging.

Another multiple frequency technique is depicted in FIG. 2, an apparatusfor using the present invention with a conductive (or active)cantilever, and the methods for its use may also be advantageous inexamining the effects of crosstalk with a view to potentiallyeliminating them in subsequent imaging. For this purpose the inventorsrefer to this apparatus and method as Dual Frequency Piezo ForceMicroscopy (DF PFM). In the DF PFM apparatus of FIG. 2 the response todriving the tip voltage of the cantilever probe, due to thepiezoelectric action acting through the contact mechanics, willtypically change as the probe is scanned over the surface. The firstsignal will then be representative of changes in the contact mechanicsbetween the tip and sample. The second signal will depend both oncontact mechanics and on the piezo electrical forces induced by thesecond excitation signal between the tip and sample. Differences betweenthe response to the first excitation and the response to the second arethus indicative of piezoelectric properties of the sample and allow thecontact mechanics to be separated from such properties.

As noted, the user often does not have independent knowledge about thedrive or damping in contact resonance. Furthermore, models may be oflimited help because they too require information not readily available.In the simple harmonic oscillator model for example, the drive amplitudeA_(drive), drive phase φ_(drive), resonant frequency ω₀ and qualityfactor Q (representative of the damping) will all vary as a function ofthe lateral tip position over the sample and may also vary in timedepending on cantilever mounting schemes or other instrumental factors.In conventional PFM, only two time averaged quantities are measured, theamplitude and the phase of the cantilever (or equivalently, the in-phaseand quadrature components). However, in dual or multiple frequencyexcitations, more measurements may be made, and this will allowadditional parameters to be extracted. In the context of the SHO model,by measuring the response at two frequencies at or near a particularresonance, it is possible to extract four model parameters. When the twofrequencies are on either side of resonance, as in the case of DFRT PFMfor example, the difference in the amplitudes provides a measure of theresonant frequency, the sum of the amplitudes provides a measure of thedrive amplitude and damping of the tip-sample interaction (or qualityfactor), the difference in the phase values provides a measure of thequality factor and the sum of the phases provides a measure of thetip-sample drive phase.

Simply put, with measurements at two different frequencies, we measurefour time averaged quantities, A₁, A₂, φ₁, φ₂ that allow us to solve forthe four unknown parameters A_(drive), φ_(drive), f₀ and Q.

FIG. 18 illustrates the usefulness of measuring the phase as a means ofseparating changes in the quality factor Q from changes in the driveamplitude A_(drive). Curve 18010 shows the amplitude response of anoscillator with a resonance frequency of f₀=320 kHz, a quality factorQ=110 and a drive amplitude A_(drive)=0.06 nm. Using DFRT PFMtechniques, the amplitude A₁ 18012 is measured at a drive frequency f₁and the amplitude A₂ 18014 is measured at a drive frequency f₂. Curve18030 shows what happens when the Q value increases to 150. The firstamplitude A₁ 18032 increases because of this increase in Q, as does thesecond amplitude A₂ 18034. Curve 18050 shows what happens when thequality factor Q, remains at 110 and the drive amplitude A_(drive)increases from 0.06 nm to 0.09 nm. Now, the amplitude measurements madeat f₁ 18052 and f₂ 18054 are exactly the same as in the case where the Qvalue increased to 150, 18032 and 18034, respectively. The amplituderesponse does not separate the difference between increasing the Q valueor increasing the drive amplitude A_(drive).

This difficulty is surmounted by measuring the phase. Curves 18020,18040 and 18060 are the phase curves corresponding to the amplitudecurves 18010, 18030 and 18050 respectively. As with the amplitudemeasurements, the phase is measured at discrete frequency values, f₁ andf₂. The phase curve 18020 remains unchanged 18060 when the driveamplitude increases from 0.06 nm to 0.09 nm. Note that the phasemeasurements 18022 and 18062 at f₁ for the curves reflecting an increasein drive amplitude but with the same quality factor are the same, as arethe phase measurements 18024 and 18064 at f₂. However, when the qualityfactor Q increases, the f₁ phase 18042 decreases and the f₂ phase 18044increases. These changes clearly separate drive amplitude changes from Qvalue changes.

In the case where the phase baseline does not change, it is possible toobtain the Q value from one of the phase measurements. However, as inthe case of PFM and thermal modulated microscopy, the phase baseline maywell change. In this case, it is advantageous to look at the differencein the two phase values. When the Q increases, this difference 18080will also increase. When the Q is unchanged, this difference 18070 isalso unchanged.

If we increase the number of frequencies beyond two, other parameterscan be evaluated such as the linearity of the response or the validityof the simple harmonic oscillator model

Once the amplitude, phase, quadrature or in-phase component is measuredat more than one frequency, there are numerous deductions that can bemade about the mechanical response of the cantilever to various forces.These deductions can be made based around a model, such as the simpleharmonic oscillator model or extended, continuous models of thecantilever or other sensor. The deductions can also be made using apurely phenomenological approach. One simple example in measuringpassive mechanical properties is that an overall change in theintegrated amplitude of the cantilever response, the response of therelevant sensor, implies a change in the damping of the sensor. Incontrast, a shift in the “center” of the amplitude in amplitude versusfrequency measurements implies that the conservative interactionsbetween the sensor and the sample have changed.

This idea can be extended to more and more frequencies for a betterestimate of the resonant behavior. It will be apparent to those skilledin the art that this represents one manner of providing a spectrum ofthe sensor response over a certain frequency range. The spectralanalysis can be either scalar or vector. This analysis has the advantagethat the speed of these measurements is quite high with respect to otherfrequency dependent excitations.

In measuring the frequency response of a sensor, it is not required toexcite the sensor with a constant, continuous signal. Other alternativessuch as so-called band excitation, pulsed excitations and others couldbe used. The only requirement is that the appropriate reference signalbe supplied to the detection means.

FIG. 16 shows one embodiment of a multi-frequency approach, with eightfrequencies f₁ through f₈ being driven. As the resonance curve changesin response to tip-surface interactions, a more complete map of thefrequency response is traced out. This may be particularly useful whenmeasuring non-linear interactions between the tip and the sample becausein that case the simple harmonic oscillator model no longer applies. Theamplitude and phase characteristics of the sensor may be significantlymore complex. As an example of this sort of measurement, one can drivethe cantilever at one or more frequencies near resonance and measure theresponse at nearby frequencies.

Scanning ion conductance microscopy, scanning electrochemicalmicroscopy, scanning tunneling microscopy, scanning spreading resistancemicroscopy and current sensitive atomic force microscopy are allexamples of localized transport measurements that make use ofalternating signals, again sometimes with an applied do bias. Electricalforce microscopy, Kelvin probe microscopy and scanning capacitancemicroscopy are other examples of measurement modes that make use ofalternating signals, sometimes with an applied dc bias. These and othertechniques known in the art can benefit greatly from excitation at morethan one frequency. Furthermore, it can also be beneficial if excitationof a mechanical parameter at one or more frequencies is combined withelectrical excitation at the same or other frequencies. The responsesdue to these various excitations can also be used in feedback loops, asis the case with Kelvin force microscopy where there is typically afeedback loop operating between a mechanical parameter of the cantileverdynamics and the tip-sample potential.

Perhaps the most popular of the AC modes is amplitude-modulated (AM)Atomic Force Microscopy (AFM), sometimes called (by Bruker Instruments)tapping mode or intermittent contact mode. Under the name “tapping mode”this AC mode was first coined by Finlan, independently discovered byGleyzes, and later commercialized by Digital Instruments.

AM AFM imaging combined with imaging of the phase, that is comparing thesignal from the cantilever oscillation to the signal from the actuatordriving the cantilever and using the difference to generate an image, isa proven, reliable and gentle imaging/measurement method with widespreadapplications. The first phase images (of a wood pulp sample) werepresented at a meeting of Microscopy and Microanalysis. Since then,phase imaging has become a mainstay in a number of AFM applicationareas, most notably in polymers where the phase channel is often capableof resolving fine structural details.

The phase response has been interpreted in terms of the mechanical andchemical properties of the sample surface. Progress has been made inquantifying energy dissipation and storage between the tip and samplewhich can be linked to specific material properties. Even with theseadvances, obtaining quantitative material or chemical properties remainsproblematic. Furthermore, with the exception of relatively soft metalssuch as In-Tn solder, phase contrast imaging has been generally limitedto softer polymeric materials, rubbers, fibrous natural materials. Onthe face of it this is somewhat puzzling since the elastic and lossmoduli of harder materials can vary over many orders of magnitude.

The present invention adapts techniques used recently in research onpolymers, referred to there as loss tangent imaging, to overcome some ofthese difficulties. Loss tangent imaging recasts our understanding ofphase imaging. Instead of understanding a phase image as depending onboth energy dissipation and energy storage, independently, we understandit as depending on an inextricable linkage of energy dissipation andenergy storage, a single term that includes both the dissipated and thestored energy of the interaction between the tip and the sample. If, forexample the dissipation increases it generally means that the storagedoes as well. This is similar to other dimensionless approaches tocharacterizing loss and storage in materials such as the coefficient ofrestitution. The loss tangent approach to materials has very earlyroots, dating back at least to the work of Zener in 1941.

In addition to loss tangent imaging, the present invention combines thequantitative and high sensitivity of simultaneous operation in afrequency modulated (FM) mode The microscope is set up for bimodalimaging with two feedback loops, the first using the first resonance ofthe cantilever and the second the second resonance. The first loop is anAM mode feedback loop that controls the tip-sample separation by keepingthe amplitude of the cantilever constant (and produces a topographicimage from the feedback signals) and at the same time compares thesignal from the cantilever oscillation to the signal from the actuatordriving the cantilever to measure changes in phase as the tip-sampleseparation is maintained constant. The second feedback loop is a FM modefeedback loop that controls the tip-sample separation by varying thedrive frequency of the cantilever. The frequency is varied in FM modethrough a phase-locked loop (PLL) that keeps the phase (again acomparison of the signal from the cantilever oscillation to the signalfrom the actuator driving the cantilever) at 90 degrees by adjusting thedrive frequency of the cantilever. A third feedback loop may beimplemented to keep the amplitude of the cantilever constant through theuse of automatic gain control (AGC). If AGC is implemented, cantileveramplitude is constant. Otherwise, if the amplitude is allowed to vary,it is termed constant excitation mode.

Much of the initial work with FM mode was in air and it has a longtradition of being applied to vacuum AFM studies (including UHV),routinely attaining atomic resolution and even atomic scale chemicalidentification. Recently there has been increasing interest in theapplication of this technique to various samples in liquid environments,including biological samples. Furthermore, FM AFM has demonstrated trueatomic resolution imaging in liquid where the low Q results in areduction in force sensitivity. One significant challenge of FM AFM hasbeen with stabilizing the feedback loops.

Briefly, when AM mode imaging with phase is combined with FM modeimaging using bimodal imaging techniques, the topographic feedbackoperates in AM mode while the second resonant mode drive frequency isadjusted to keep the phase at 90 degrees. With this approach, frequencyfeedback on the second resonant mode and topographic feedback on thefirst are decoupled, allowing much more stable, robust operation. The FMimage returns a quantitative value of the frequency shift that in turndepends on the sample stiffness and can be applied to a variety ofphysical models.

Bimodal imaging involves using more than one resonant vibrational modeof the cantilever simultaneously. A number of multifrequency AFM schemeshave been proposed to improve high resolution imaging, contrast, andquantitative mapping of material properties, some of which have alreadybeen discussed above.

With bimodal imaging the resonant modes can be treated as independent“channels”, with each having separate observables, generally theamplitude and phase. The cantilever is driven at two flexuralresonances, typically the first two, as has been described above. Theresponse of the cantilever at the two resonances is measured and used indifferent ways as shown in FIG. 20. It will be noted that the FIG. 20apparatus bears some resemblance to the apparatus shown in FIG. 1.

FIG. 20 is a block diagram of a preferred embodiment of an apparatus forprobing two flexural resonances of a cantilever in accordance with thepresent invention. The sample 1010 is positioned below the cantileverprobe 1020. The chip of the cantilever probe 1030 is driven by amechanical actuator 1040, preferably a piezoelectric actuator, but othermethods to induce cantilever motion known to those versed in the artcould also be used. The motion 1150 of the cantilever 1020 relative tothe frame of the microscope (not shown) is measured with a detector (notshown), which could be an optical lever or another method known to thoseversed in the art. The cantilever chip 1030 is moved relative to thesample 1010 by a scanning apparatus (not shown), preferably apiezo/flexure combination, but other methods known to those versed inthe art could also be used.

The motion imparted to the cantilever chip 1030 by actuator 1040 iscontrolled by excitation electronics that include at least two frequencysynthesizers 1080 and 1090. The signals from these frequencysynthesizers could be summed together by an analog circuit element 1100or, preferably, a digital circuit element that performs the samefunction. The two frequency synthesizers 1080 and 1090 provide referencesignals to lockin amplifiers 1110 and 1120, respectively. As with otherelectronic components in this apparatus, the lockin amplifiers 1110 and1120 can be made with analog circuitry or with digital circuitry or ahybrid of both. For a digital lockin amplifier, one interesting andattractive feature is that the lockin analysis can be performed on thesame data stream for both flexural resonances. This implies that thesame position sensitive detector and analog to digital converter can beused to extract information at the two distinct resonances.

Resonance 1:

As shown in the upper shaded area of FIG. 20, the flexural resonancesignal from frequency synthesizer 1080 is compared to the cantileverdeflection signal 1150 through lockin amplifier 1110. This feedback loopcontrols the z actuator (not shown) which moves the cantilever chip 1030relative to the sample 1010 and thus controls the amplitude of thecantilever 1020 and the tip-sample separation. The amplitude signalresulting from this feedback is used to create a topographic image ofthe sample 1010. Simultaneously, the phase of the cantilever 1020 iscalculated from this comparison and together with the amplitude signalis used to generate the tip-sample loss tangent image.

Resonance 2:

As shown in the lower shaded area of FIG. 20, the flexural resonancesignal from frequency synthesizer 1090 is compared to the cantileverdeflection signal 1150 through lockin amplifier 1120 and a second phaseof the cantilever probe 1020 is calculated from this comparison. ThePhase Locked Loop (PLL) device 1160 in turn maintains this phase at 90degrees by making appropriate adjustments in the flexural resonancesignal from frequency synthesizer 1090. The required adjustment providesa FM based measure of tip-sample stiffness and dissipation. Tip-samplestiffness and dissipation can also be measured from the amplitude andphase of the flexural resonance signal from frequency synthesizer 1090.FM mode may also employ an AGC device to maintain the amplitude of thecantilever 1020 at a constant value.

The foregoing bimodal imaging approach to quantitative measurements withLoss Tangent and AM/FM imaging techniques has the great advantage ofstability. With topographic feedback confined to the first resonant modeand FM mode to the second resonant mode, even if the PLL or AGC controlloops become unstable and oscillate, there is little or no effect on theability of the first mode to stably track the surface topography.

In order to highlight some important limitations it is useful to take amathematical approach to Loss Tangent imaging. As already noted in AMAFM operation, the amplitude of the first resonant mode is used tomaintain the tip-sample distance. The control voltage, typically appliedto a z-actuator results in a topographic image of the sample surface. Atthe same time, the phase of the first resonant mode will vary inresponse to the tip-sample interaction. This phase reflects bothdissipative and conservative interactions. A tip which indents a surfacewill both dissipate viscous energy and store elastic energy—the twoenergies are inextricably linked. The loss tangent is a dimensionlessparameter which measures the ratio of energy dissipated to energy storedin one cycle of a periodic deformation. The loss tangent of thetip-sample interaction can be described by the following relationinvolving the measured cantilever amplitude V and phase φ:

${\tan\;\delta} = {\frac{G^{''}}{G^{\prime}} = {{\frac{\left( {F_{ts} \cdot \overset{.}{z}} \right)}{\omega\left( {F_{ts} - z} \right)} \approx \frac{{\frac{V}{V_{free}}\frac{\omega}{\omega_{free}}} - {\sin\;\phi}}{{\cos\;\phi} - {Q\frac{V}{V_{free}}\left( {1 - \frac{\omega^{1}}{\omega_{free}^{2\;}}} \right)}}} = {\frac{{\Omega\;\alpha} - {\sin\;\phi}}{{Q\;{\alpha\left( {1 - \Omega^{2}} \right)}} - {\cos\;\phi}}.\mspace{31mu}({FullTand})}}}$

In this expression, F_(tz) is the tip-sample interaction force, z is thetip motion, ż is the tip velocity, ω is the angular frequency at whichthe cantilever is driven and

represents a time-average. The parameter V_(free) is the “free” resonantamplitude of the first mode, measured at a reference position and is animportant calibration parameter. Note that because the amplitudes appearas ratios in the FullTand equation, they can be either calibrated oruncalibrated in terms of the optical detector sensitivity. In the finalexpression of FullTand we have defined the ratios Ω≡ω/ω_(free) andα≡A/A_(free)=V/V_(free). If we operate on resonance (Ω=1), theexpression can be simplified to:

${\tan\;\delta} = {\frac{\left( {F_{ts} \cdot \overset{.}{z}} \right)}{\omega\left( {F_{ts} \cdot z} \right)} \approx {\frac{{\sin\;\phi} - \alpha}{\cos\;\phi}.\mspace{31mu}({SimpleTand})}}$There are some important implications of these equations:

1. Attractive interactions between the tip and the sample will ingeneral make the elastic denominator ω

F_(n)·z

of equations FullTand and SimpleTand smaller. This will increase thecantilever loss tangent and therefore over-estimate the sample losstangent.

2. Tip-sample damping with origins other than the sample loss modulus,originating from interactions between, for example, a water layer oneither the tip or the sample will increase the denominator in equationsFullTand and SimpleTand.

These factors point out an important limitation of loss tangent imaging.Equations equations FullTand and SimpleTand are really the loss tangentof the cantilever—but not necessarily the loss tangent originating fromthe sample mechanics: G″ and G′. With proper choice of operatingparameters, this effect can be minimized, improving the estimate of theloss tangent. For example, in the case of the mechanical loss tangent ofa polymer surface, the scan should be in repulsive mode so that thecantilever is sampling the short range repulsive forces controlled bythe sample elastic and loss moduli. It is important to take the stepsrequired for this to be so.

To understand the importance of these parameters and to understand thepractical limitations of loss tangent imaging, it is useful to performan error analysis on the measurement. Errors in the measured losstangent depend on phase errors and amplitude errors. Using standarderror analysis, the fractional loss tangent error which is dependent onuncertainties in the amplitude and phase of the measurement is given by

$\frac{\Delta tan\delta}{\tan\;\delta} = {{{\frac{1}{\cos\;{\varphi \cdot \tan}\;\delta}}\Delta\; V_{r}} + {{{{\tan\;\varphi} - 1}}{\Delta\varphi}}}$

For a simple harmonic oscillator cantilever model, there is a monotonicrelationship between the phase and the drive frequency,

${\tan\;\varphi} = {\frac{{\omega\omega}_{free}/Q}{\omega_{free}^{2} - \omega^{2\;}}.}$This implies that measurements of the resonant frequency are equivalentto measuring a frequency-dependent phase shift φ(ω) subject to thecondition φ(ω=ω_(free))=90°. The foregoing equation can be plotted forvarious experimental situations.

In the two decade history of phase imaging, there are very few examplesof phase contrast over relatively hard materials. The above erroranalysis along with the Ashby (1987) provides some insight into this.There is a general trend that less elastic materials tend to be morelossy. However, there are many examples where a stiffer material mightalso exhibit higher dissipation. This underscores the danger in simplyinterpreting phase contrast in terms of only the sample elasticity.

Thermal noise limits the loss tangent resolution at small values and athigh values. In particular, since the loss tangent diverges at acantilever phase of 90 degrees, fluctuations near this point have a verystrong effect on the estimated loss tangent.

This observation is remarkably consistent with a literature search forphase imaging. There are many examples of phase imaging of polymericmaterials and very few of metals and ceramics with tan δ<10⁻²,consistent with the error analysis above. This insight is one benefit ofthe loss tangent point of view in that it provides some insight into ageneral contrast limitation the AFM community has been subject to for along time.

It may be useful to mention here some practical experimental limitationson loss tangent imaging. Proper choice of the zero-dissipation point iscritical for proper calibration of the tip-sample dissipation in theloss tangent mode. In particular, squeeze film damping can have a strongeffect on the measured dissipation. Squeeze film damping causes thecantilever damping to increase as the body of the cantilever moves closeto the sample surface. For rough surfaces, this can mean that thecantilever body height changes with respect to the average sampleposition enough to cause crosstalk artifacts in the measured dissipationand therefore the loss tangent. An example is shown in FIG. 21 of aSilicon substrate patterned with SU8, an epoxy commonly used forphotolithography. The silicon is expected to have a high modulus ˜150GPa while the literature puts the modulus of the SU8 at Lorenz et al.reported that SU-8 has a modulus of elasticity of 4.02 GPa. Since theSU8 is a polymer, we expect it to have some viscoelastic properties; inany event more than that of the silicon. As shown in the figure, the SU8layer was >1.5 um thick. Surprisingly however, the loss modulus ascalculated using the FullTand equation shown in FIG. 3(b) indicates thatthe tan δ_(Si)>tan δ_(SU8). This non-physical result can be explained byair damping differences over the two materials. Since the Si is ˜1.5 umlower than the SU8 epoxy, when the tip is measuring the Si, itexperiences increased air damping. This is interpreted as a larger losstangent over the Si.

FIG. 21 shows the topography 2101 of a Si-epoxy (SU8) patterned wafer.The loss tangent 2102 calculated using the FullTand equation showssurprising contrast inversion: the loss tangent of the cantilever overthe Si is higher than over the relatively lossy Su8 polymer. Thecorrected loss tangent 2103 using the FullTand equation and theschematic of the sample shows the ˜1.5 um thick SU8 epoxy step and theamplitude curves measured over the Si (red) and SU8 (black). The stepsused to acquire the corrected loss tangent image 2103 are described inthe text.

The reference amplitudes 2105 and 2106 for the cantilever are differentover the two regions, because of the large difference in sample height.As discussed above, these reference values are critical for correctlyestimating the loss tangent of the sample. Measuring the referenceamplitude at the same height above the sample should mitigate thissystematic error. Qualitatively, the results shown in 2103 areconsistent with this conclusion—the dissipation over the lower Sifeatures appears larger than over the SU8, consistent with air dampingplaying a role in miscalibrating the reference amplitude.

To correct for this topographic crosstalk, we have implemented apixel-by-pixel referencing method for loss tangent imaging. In thismethod, illustrated in FIG. 22, where a normal AM mode imaging scan 2201is made while the topography, amplitude and phase are recorded. Thecantilever is then raised a height (Δz) above the surface and the samex-y scan is repeated. During this second pass 2202, the cantilever isoperated in a phase-locked loop, allowing the resonant frequency andreference amplitude at that frequency to be measured. The drivefrequency and quality factor 2200 are measured in the initial cantilevertune, typically performed before or after the scanning steps.

FIG. 22 shows steps to calculate the corrected loss tangent.

1. 2203 first the drive frequency is chosen and the quality factor areestimated in a tune far away from the surface.

2. 2202 the reference amplitude and resonant frequency are measured byscanning the cantilever a preprogrammed height above the surface.

3. 2201 the amplitude and phase during normal tapping mode are measuredon the surface.

4. The parameters measured in steps 2203, 2202 and 2201 are used tocalculate the loss tangent on a pixel by pixel basis. The referenceheight at each x-y pixel location is set by a single delta height (Δz)parameter—which in turn determines the reference amplitude and resonancefrequency.

The results of this approach are shown in FIG. 21. After applying thecorrection 2103, the loss tangent over the Si is on the order of 0.01and over the SU8 on the order of 0.05. The most important conclusion isthat the order is now correct for the two materials—Si is smaller andSU8 is larger. Furthermore, the values over Si are consistent with thelimits imposed by the noise analysis shown in the equation above.

Another improvement in loss tangent imaging is to include energy beingtransferred to higher harmonics of the cantilever. This can be asignificant effect at low Q values. Energy losses to higher harmonics ofthe cantilever are more significant at lower Q than at higher Q. Tamayohas accounted for this energy dissipation by including the harmonicresponse of the cantilever. By extending this analysis to the storagepower, we derived an expression for the loss tangent of the equationabove that now includes harmonic correction terms:

${\tan\;\delta} = {\frac{{\sin\;\phi_{1}} - {\sum\limits_{n \geq 1}^{N}{n^{2}\frac{A_{n}^{2}}{A_{1}A_{free}}}}}{{\cos\;\phi_{1}} - {Q{\sum\limits_{n \geq 1}^{N}{\left( {n^{2} - 1} \right)\frac{A_{n}^{2}}{A_{1}A_{free}}}}}}.\mspace{31mu}({HarmTand})}$

In equation HarmTand, n is the order of the harmonic (ranging from thefundamental at n=1 up to the limit N) and A_(n) the amplitude at then_(th) harmonic. In the case of the dissipation (the numerator), theharmonics behave as a “channel” for increased damping. Specifically, ifenergy goes into the harmonics, the fundamental mode, damping willappear to increase. In the case of the storage term (the denominator),energy going into the harmonics looks like a reduction in the kineticenergy of the cantilever. This has the effect of reducing the apparentstorage power in equations SimpleHarm and the equation above HarmTand.These two effects act in concert to increase the measured loss tangent.

In addition to measuring many of the harmonics of the loss tangentfundamental, the error associated with harmonic loss can be estimatedand improved upon by simply measuring the response of the cantilever atone harmonic, for example the 6th or 4th harmonic, that is a harmonicclose to the next highest resonant mode.

FM AFM has become a powerful technique for imaging surfaces at atomicresolution, and manipulating atomic surfaces. By measuring the frequencyshift as the tip interacts with the surface, it is possible to quantifytip-sample interactions. In particular, the frequency shift of acantilever in FM mode is given by the equation

${\Delta\; f_{2}} = {{f_{0,2}\frac{\left\langle {F_{ts}z} \right\rangle}{k_{2}A_{2}^{2}}} \approx {\frac{f_{0,2}}{2}{\frac{k_{ts}}{k_{2}}.}}}$In addition to the terms described for the FullTand equation, f_(0,2) isthe second resonance frequency measured at a “free” or referenceposition, Δf₂ is the shift of the second resonant mode as the tipinteracts with the surface, k₂ is the stiffness of the second mode andA₂ is the amplitude of the second mode as it interacts with the surface.As with the expression for the loss tangent, it does not directlyinvolve the optical lever sensitivity. Thus, we can relate the measuredfrequency shift to an equation for tip-sample stiffness:

$k_{ts} \approx {\frac{2k_{2}\Delta\; f_{2}}{f_{0,2}}.}$

The second mode resonant behavior provides a direct measure of thetip-sample interaction forces. FIG. 23 shows an example of this. In thismeasurement, the amplitude of the first resonance 2301 is plotted versustip-sample distance. As the cantilever approaches the surface, it firstexperiences net attractive forces which reduce the amplitude 2302. At acertain point, there is a switch from net attractive to net repulsiveinteractions which results in a discontinuity in the amplitude-distancecurve 2303. This effect is well-known in the literature. After thatpoint, the interaction is dominated by repulsive forces and the curve issomewhat different 2304. The phase versus distance curve 2305 measuredsimultaneously gives additional information. The phase shift is positiveduring the net-attractive interaction 2306 portion of the approachcurve. The attractive-repulsive transition 2307 shows the switch betweenthe attractive dominated portion 2306 and the repulsive dominated 2308portion of the curve. Note that each transition 2303 and 2307 have somesmall hysteresis between the approach and retract portions of thecurves. The families of second resonant mode phase curves 2309, 2310 andsecond resonant mode amplitude curves 2311 and 2312 were measured whilethe first mode was held constant by a z-feedback loop. The feedback loopwas enabled, keeping the first mode at an essentially constant value byadjusting the z-height of the cantilever while the drive frequencyoperating on the second mode 1090 was ramped. The frequency tunes weremade by sweeping the second drive frequency 1090 but could bealternatively be chirped, band-excited, intermodulated or excited insome other manner that explores the frequency content of theinteraction. The amplitude tune far from the surface 2315 and the phasefar from the surface 2314 are plotted as dashed lines. During theattractive portion of the interaction, the phase curves 2309 move to theleft, indicating that the resonance has shifted to a lower frequency, asexpected from an attractive interaction. During the repulsive portion ofthe interaction, the phase curves move to the right, shifting towardshigher frequencies 2310 as expected from a stiffer interaction. Theamplitude curves show corresponding behavior, with the peak shifting tothe left (lower frequencies) during the attractive portion of theinteraction 2311 and towards higher frequencies during the repulsiveportion of the interaction 2312. In addition, the peak values of theamplitude curves decreases, indicating an overall increase in tip-sampledamping or dissipation 2311, 2312. By engaging a phase locked loop (PLL)1160 and repeating the amplitude ad phase versus distance experiment,the output of the f₂ 2010 and A₂ 2013 results in the curve 2313. This isnotable in that it tracks the peak values of the curves in the A₂ vsfrequency display in FIG. 23. This implies that the PLL can be used, atleast in this case as a rapid means of interrogating the cantilever asto its second mode response. Note that other methods for measuring thiscould also be used as discussed above including the DART, BandExcitation, chirping, intermodulation and other techniques that provideinformation about the frequency response.

In some cases, it is advantageous to omit the phase-locked loop and tosimply measure the phase of the second mode. Examples include when thetransfer function of the cantilever actuation mechanism is subject tofrequency-dependent amplitude and phase shifts. This is essentiallybimodal or DualAC mode. This is often the case for operation in fluid,but can even be the case for higher quality factor situations where thecantilever actuator has a frequency-dependent transfer function. Thiscan be used (as can the other modes described here) with a large varietyof cantilever actuation means including acoustic, ultrasonic, magnetic,electric, photothermal, photo-pressure and other means known in the art.

To relate the phase shift to the stiffness of the sample, we can startwith the relationship between the frequency and phase shifts for asimple harmonic oscillator;

${\frac{\partial\varphi}{\partial x} = \frac{\left( {x^{2} + 1} \right)Q}{{\left( {x^{2} - 1} \right)^{2}Q^{2}} + x^{2}}},$where x≡f_(drive,2)/f_(0,2) is the ratio of the drive frequency to theresonant frequency measured at the reference position. Using theSimpleTand equation, this can be manipulated to give the tip-sampleinteraction stiffness in terms of the phase shift measured at a fixeddrive frequency:

$k_{ts} \approx {{- \frac{f_{0,2}}{f_{{drive},2}}}\frac{\Delta\varphi}{Q}{k^{2}.}}$

Note that this expression is only valid for small frequency shifts.While the complete, nonlinear expression is analytic, our experience isthat the frequency shift of the higher modes are typically a few partsin a thousand or smaller, justifying the simple expression.

This mode of operation can be extremely sensitive, down to the level ofrepeating images of single atomic defects as shown in FIG. 26. In thisFigure, two repeating images 2601, 2602 of the surface of Calcite influid are shown, each with the same defect 2603, 2604 visible. Thecorresponding stiffness images are shown on the right 2605, 2606 withthe single atomic stiffness defect 2607, 2608 visible in each image.Line sections taken across the stiffness images show the sub-nanometerpeak in the stiffness associated with the defect in each image 2609.This very high resolution mechanical property measurement in fluidrepresents a very significant advance in the materials propertiesmeasurements using AFM.

Since loss tangent can be measured using the first mode and FM ismeasured using the second resonance mode, both measurements can be madesimultaneously. There are some practical experimental conditions toconsider when applying this technique to nano-mechanical materialsproperties measurements:

The tip is sensitive to G′ and G″ only in repulsive mode. Repulsive modeis favored for:

1. larger cantilever amplitudes (>1 nm)

2. stiffer cantilevers (>1 N/m)

3. sharp tips and

4. lower setpoints (typically 50% of the free amplitude).

As a check, the first mode phase should always be <90° and typically<50° for the majority of materials. This assures you are sampling therepulsive tip-sample interactions. Good feedback tracking (avoidparachuting, make sure trace and retrace match) assures good sampling ofthe mechanical properties. Finally, the accuracy of both techniquesdepends strongly on careful tuning of the cantilever resonances.Specifically, the resonances should be <10 Hz error and the phase shouldbe within 0.5 degrees. These are more stringent conditions than usualfor AM mode but are well within the capabilities of commercial AFMs,with proper operation.

FIG. 25 shows an example of simultaneous loss tangent 2501 and stiffness2502 mapping of an elastomer-epoxy sandwich. A natural rubber sheet wasbonded to a latex rubber sheet with an epoxy. The sandwich was thenmicrocryotomed and imaged. From macroscopic measurements, the elasticityE˜40 MPa (natural rubber):4 GPa (epoxy):43 MPa (latex rubber) measuredwith a Shore durometer, while the macroscopic loss tangents wereestimated to be 1.5, 0.1:2 and 2, respectively, measured with a simpledrop test. An AC160 cantilever with a fundamental resonance of 310 kHzand a second mode resonance of 1.75 MHz was used. Histograms of the losstangent 2503 and of the stiffness 2504 show clear separation of thethree components. Note that the surface roughness was on the order of500 nm and despite this, the materials are still clearly differentiated.

Since the second mode resonance depends on the interaction stiffnessk_(is), the material modulus can be mapped by applying a particularmechanical model. One of the most simple models is a Hertz indenter inthe shape of a punch. In this case, the elasticity of the sample isrelated to the tip-sample stiffness by the relation k_(is)=2E′a, where ais a constant contact area. Combining this with the SimpleTand equationresults in the expression

$E^{\prime} = {\frac{\Delta\; f_{2}}{f_{0,2}}{\frac{k_{2}}{a}.}}$

Thus if the contact radius and spring constant are known, the samplemodulus can be calculated. Of course, other tip shapes could be used inthe model. Calibration of the tip shape is a well-known problem.However, it is possible to use a calibration sample that circumventsthis process. As a first step, we have used a NIST-traceable ultra highmolecular weight high density polyethelene (UHMWPE) sample to firstcalibrate the response of the AC160 cantilever. The above equation canbe rewritten as E′=C₂Δf₂, where C₂ is a constant, measured over theUHMWPE reference that relates the frequency shift to the elasticmodulus. This can then be applied to unknown samples.

Finally, this technique can be performed at high speeds using smallcantilevers. The response bandwidth of the f^(th) resonant mode of acantilever is BW_(i)=πf_(i,0)/Q_(i) where f_(i,0) is the resonantfrequency of the i^(th) mode and Q_(i) is the quality factor. Toincrease the resonance frequency without changing the spring constantcan be done by making cantilevers smaller. In contrast to normal AMimaging, the second resonant mode must still be accessible to thephotodetector, requiring f_(2,0)<10 MHz for the Cypher AFM of AsylumResearch Corporation. An example is shown in FIG. 24, where a EPDH/Epoxycryo-microtomed boundary is measured at a 2 Hz 2401 and 20 Hz 2402 linescan rates. These images were 5 um on a side and were acquired with anAC55 cantilever from Olympus (f_(1,0)≈1.3 MHz, f₁₂₀≈5.3 MHz). Thehistograms of the 2 Hz 2403 and 20 Hz 2404 images show very littledeviation between the peak elasticity measurements indicating the highspeed image acquisition did not significantly affect the results.

Loss tangent and AM-FM provide two additional tools for quantifyingnanoscale mechanical properties. These modes are compatible withconventional AM imaging, meaning that high resolution, high speedmechanical properties can be made on an enormous variety of samples:

-   -   1. No invols calibration—instead relies on automated frequency        tunes        -   a. Sader k calibration        -   b. Ratios of amplitudes        -   c. Resonant frequency shifts    -   2. Based on reliable, proven and well understood technology and        physics—not a mysterious black box        -   a. Tapping mode    -   3. Simple optimization        -   a. Decoupled loops            -   i. Resonance 1: Topography and loss tangent            -   ii. Resonance 2: Stiffness and Dissipation    -   4. Quantitative results on a wide variety of samples    -   5. High speed        -   a. Demonstrated 10-20 Hz linescan rates on rough samples    -   6. Extremely high resolution        -   a. Repeating single atomic defect stiffness resolution

In general, one can choose to use any higher resonant mode for stiffnessmapping. There are a couple of things to consider when making thechoice.

-   -   1. Avoid modes that are at or very close to integer multiples of        the first resonance. If they are at integer values, one gets        harmonic mixing between the modes which can cause instabilities.        For example, it is quite common that the second resonance of an        AC240 is ˜6× the first. For that reason, we often use the third        resonance mode instead, typically ˜15.5× the first.    -   2. The sensitivity is optimized when the stiffness of the mode        is tuned to the tip-sample stiffness.

Note that caveat 1 above applies to AFM measurement modes where thecantilever is not necessarily being driven at the second resonant mode.If a higher mode is too close to an integer multiple of the drivefrequency, unwanted harmonic coupling can take place that leads tospurious, noisy and/or difficult to interpret results.

FIG. 27 shows the effects of choosing the resonant mode that was softer,matched or stiffer than the tip-sample stiffness, as mentioned in caveat2 above. In this work, the fundamental resonance of an AC200 cantileverwas at ˜1.15 kHz. The second resonance 2701 was at ˜500 kHz, the third2702 was at ˜1.4 MHz, the fourth 2703 at ˜2.7 MHz, the fifth 2704 at˜4.3 MHz and the sixth 2705 was at ˜6.4 MHz. The elasticity images,measured and calculated as discussed above are shown 2706, 2707, 2708,2709 and 2710 in ascending order. Note that the contrast in the secondmode (soft) 2706 is relatively low, as is the contrast in the 5^(th) and6^(th) mode images (stiff) 2709, 2710. This is explicitly visible in theelasticity histograms 2711, 2712, 2713, 2714 and 2715, arranged inascending order.

Note that to optimize the response of the cantilever to mechanicalstiffness contrast and accuracy, it may be advantageous to tune theamplitude of the second mode so that it is large enough to be above thedetection noise floor of the instrument, but small enough to notsignificantly affect the trajectory and behavior of the fundamental modemotion as discussed above and in reference to FIG. 4. To this end, itmay be useful to plot the higher mode amplitude and/or phase and/orfrequency as a function of the second mode drive amplitude. This can bedone for example, while the first mode is employed in a feedback loopcontrolling the tip-sample separation or in a pre-determined tip-sampleposition. By measuring A1 4010, 4050, P1 4020, 4060 and P2 4040 as afunction of the higher mode drive amplitude 1090, the user can chose adrive amplitude that optimizes the signal to noise while minimizing theeffect of nonlinearities on the measured signals.

Another issue with making the stiffness and other measurementsquantitative is quantification of the higher mode stiffness. In general,this is a challenging measurement. One method is to extend the thermalnoise measurement method to higher modes as indicated in FIG. 28. Thethermal measurement depends on accurately measuring the optical leversensitivity. This can be done by driving each resonant mode separatelyand then measuring the slope of the amplitude-distance curve as thecantilever approaches a hard surface. This calibrated opticalsensitivity can then be used in a thermal fit as is well known in theart to get the spring constant for that particular resonant mode.Typical fits for the fundamental 2801, second resonance 2802 and thirdresonance 2803 yielded optical lever sensitivities that were then usedto fit the thermal noise spectra to extract the spring constants for thefirst 2804 second 2805 and third mode 2806 of the AC240 cantilever.

The described embodiments of the invention are only considered to bepreferred and illustrative of the inventive concept. The scope of theinvention is not to be restricted to such embodiments. Various andnumerous other arrangements may be devised by one skilled in the artwithout departing from the spirit and scope of the invention.

What is claimed is:
 1. An atomic force microscope, comprising: an atomicforce microscope cantilever, having a base and a probe tip; a surfacefor holding a sample to be measured by the cantilever, adjacent to thecantilever; a controller; and a cantilever driving element, driven bysaid controller, the controller receiving a signal indicative ofcantilever motion, the controller exciting the cantilever at both offirst and second eigenmodes, including one of the eigenmodes thatprovides attractive interactions between the cantilever and sample, andanother of the eigenmodes that provides repulsive interactions betweenthe cantilever and sample, the controller exciting the cantilever at thefirst and second eigenmodes while controlling a distance between thecantilever and the sample, and the controller measuring the amplitudeand/or phase of the cantilever at least at one of first and secondfrequencies associated with one of the first and second eigenmodes. 2.The microscope as in claim 1, wherein the controller measures amplitudeand/or phase at both of the first and second frequencies.
 3. Themicroscope as in claim 1, wherein the controller measures amplitudeand/or phase at one of the frequencies, and keeps constant amplitudeand/or phase of the other of the frequencies.
 4. The microscope as inclaim 1, wherein said first frequency is a resonant frequency of thecantilever.
 5. The microscope as in claim 1, wherein said controller isconnected into a feedback loop, and maintain the probe tip of thecantilever in a pre-established relationship with respect to the surfaceof the sample in a Z axis direction defined between the probe tip andthe sample, while scanning the sample.
 6. The microscope as in claim 5,wherein said first eigenmode is at a first resonant frequency of thecantilever, wherein said probe tip oscillates at said resonantfrequency, and said resonant frequency is applied to said feedback loopto control interactions between the cantilever and sample to maintain anamplitude of oscillation at the first resonant frequency of the probetip essentially constant at an amplitude setpoint.
 7. The microscope asin claim 5, wherein said second eigenmode is at a second resonantfrequency of the cantilever.
 8. The microscope as in claim 1, whereinthe measured amplitude and/or phase is used to determine informationabout the sample surface.
 9. The microscope as in claim 1, wherein thefirst eigenmode excites in a first mode, and the second eigenmodeexcites in a second mode, and in the absence of any second modeexcitation, the first eigenmode is interacting with the surface in theattractive mode, and after the second mode is added, the first eigenmodeis in a repulsive mode, induced by the addition of the second eigenmodeenergy.
 10. A method of operating an atomic force microscope,comprising: exciting an atomic force microscope cantilever, having abase and a probe tip relative to a surface for holding a sample to bemeasured by the cantilever, adjacent to the cantilever; and establishinga feedback loop between the cantilever and a controller that carries outthe exciting, the controller receiving a signal indicative of cantilevermotion, said exciting being at both of first and second eigenmodes,including one of the eigenmodes that provides attractive interactionsbetween the cantilever and sample, and another of the eigenmodes thatprovides repulsive interactions between the cantilever and sample, saidexciting also controlling a distance between the cantilever and thesample, and measuring the amplitude and/or phase of the cantilever atleast at one of first and second frequencies associated with one of thefirst and second eigenmodes.
 11. The method as in claim 10, wherein themeasuring comprises measuring amplitude and/or phase at both of thefirst and second frequencies.
 12. The method as in claim 10, wherein themeasuring measures amplitude and/or phase at one of the frequencies, andkeeps constant amplitude and/or phase of the other of the frequencies.13. The method as in claim 10, wherein said first eigenmode is at afirst resonant frequency of the cantilever, wherein said probe tiposcillates at said resonant frequency, and said resonant frequency isapplied to said feedback loop to control interactions between thecantilever and sample to maintain an amplitude of oscillation at thefirst resonant frequency of the probe tip essentially constant at anamplitude setpoint.
 14. The method as in claim 10, wherein the firsteigenmode excites in a first mode, and the second eigenmode excites in asecond mode, and in the absence of any second mode excitation, the firsteigenmode is interacting with the surface in the attractive mode, andafter the second mode is added, the first eigenmode is in a repulsivemode, induced by the addition of the second eigenmode energy.